7.9.1 problem 13

Internal problem ID [249]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 2. Linear Equations of Higher Order. Section 2.2 (General solutions of linear equations). Problems at page 122
Problem number : 13
Date solved : Tuesday, March 04, 2025 at 11:06:20 AM
CAS classification : [[_3rd_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime }+2 y^{\prime \prime }-y^{\prime }-2 y&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=2\\ y^{\prime \prime }\left (0\right )&=0 \end{align*}

✓ Maple. Time used: 0.013 (sec). Leaf size: 18

ode:=diff(diff(diff(y(x),x),x),x)+2*diff(diff(y(x),x),x)-diff(y(x),x)-2*y(x) = 0; 
ic:=y(0) = 1, D(y)(0) = 2, (D@@2)(y)(0) = 0; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = \frac {\left (4 \,{\mathrm e}^{3 x}-1\right ) {\mathrm e}^{-2 x}}{3} \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 22

ode=D[y[x],{x,3}]+2*D[y[x],{x,2}]-D[y[x],x]-2*y[x]==0; 
ic={y[0]==1,Derivative[1][y][0] ==2,Derivative[2][y][0] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {4 e^x}{3}-\frac {1}{3} e^{-2 x} \]

✓ Sympy. Time used: 0.209 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-2*y(x) - Derivative(y(x), x) + 2*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 3)),0) 
ics = {y(0): 1, Subs(Derivative(y(x), x), x, 0): 2, Subs(Derivative(y(x), (x, 2)), x, 0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {4 e^{x}}{3} - \frac {e^{- 2 x}}{3} \]